Growth of associated monomial algebras with application to Manturov groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915715595894784 |
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| author | Zhao, Xiangui |
| author_facet | Zhao, Xiangui |
| contents | It is well-known that an associative algebra shares the same growth and Gelfand-Kirillov dimension (GK-dimension) as its associated monomial algebra with respect to a degree-lexicographic order. This article mainly investigates the relationship between the GK-dimension of an algebra and that of its associated monomial algebra with respect to a monomial order. We obtain sufficient conditions on a monomial order such that these two algebras have the same GK-dimension. Our result generalizes the well-known result and has several applications. In particular, as an application, we study the growth of Manturov $(k,n)$-groups for positive integers $n>k$. It is shown that the Manturov $(1,n)$-group has growth equal to $0$ for all $n>1$; the Manturov $(2,3)$-group has growth equal to $2$; and, for all $n>k\geq3$, the Manturov $(k,n)$-group contains a free subgroup of rank $2$ and thus has exponential growth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_04477 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Growth of associated monomial algebras with application to Manturov groups Zhao, Xiangui Rings and Algebras Group Theory 16P90, 16Z10, 20F05, 20F10 It is well-known that an associative algebra shares the same growth and Gelfand-Kirillov dimension (GK-dimension) as its associated monomial algebra with respect to a degree-lexicographic order. This article mainly investigates the relationship between the GK-dimension of an algebra and that of its associated monomial algebra with respect to a monomial order. We obtain sufficient conditions on a monomial order such that these two algebras have the same GK-dimension. Our result generalizes the well-known result and has several applications. In particular, as an application, we study the growth of Manturov $(k,n)$-groups for positive integers $n>k$. It is shown that the Manturov $(1,n)$-group has growth equal to $0$ for all $n>1$; the Manturov $(2,3)$-group has growth equal to $2$; and, for all $n>k\geq3$, the Manturov $(k,n)$-group contains a free subgroup of rank $2$ and thus has exponential growth. |
| title | Growth of associated monomial algebras with application to Manturov groups |
| topic | Rings and Algebras Group Theory 16P90, 16Z10, 20F05, 20F10 |
| url | https://arxiv.org/abs/2601.04477 |